Multiscale Computations for Highly Oscillatory Problems

We review a selection of essential techniques for constructing computational multiscale methods for highly oscillatory ODEs. Contrary to the typical approaches that attempt to enlarge the stability region for specialized problems, these lecture notes emphasize how multiscale properties of highly oscillatory systems can be characterized and approximated in a truly multiscale fashion similar to the settings of averaging and homogenization. Essential concepts such as resonance, fast-slow scale interactions, averaging, and techniques for transformations to non-stiff forms are discussed in an elementary manner so that the materials can be easily accessible to beginning graduate students in applied mathematics or computational sciences.

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